Digital SAT Math
SAT Math: Find the Radius of Circle Defined by x² + y² - 14x + 10y = 7
• Direct Answer / Key Takeaway
(x - 7)^2 + (y + 5)^2 = 7 + 49 + 25 = 81 => r = sqrt(81) = 9.
Exact Question Text
The equation x^2 + y^2 - 14x + 10y = 7 defines a circle in the xy-plane. What is the radius of the circle?
1. Step-by-Step Formal Solution
- Group: (x^2 - 14x + 49) + (y^2 + 10y + 25) = 7 + 49 + 25 = 81.
- (x - 7)^2 + (y + 5)^2 = 81 => r = 9.
2. Fast Desmos / Calculator Shortcut
Type x^2 + y^2 - 14x + 10y = 7 in Desmos. Measure diameter (18) / 2 = 9.
3. Distractor Trap Analysis
- Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
- Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
- Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
- Option Correct: Accurately satisfies all algebraic and logical constraints.
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